Existence and uniqueness of solution to a hyperbolic-parabolic free boundary problem for biofilm growth
Abstract
This work presents the existence and uniqueness of solution to a free boundary value problem related to biofilm growth. The problem consists of a system of nonlinear hyperbolic partial differential equations governing the microbial species growth, and a system of parabolic partial differential equations describing the substrate dynamics. The free boundary evolution is governed by an ordinary differential equation that accounts for the thickness of the biofilm. We use the method of characteristics and fixed point strategies to prove the existence and uniqueness theorem in small and all times. All the equations are converted into integral equations, in particular this transformation is made for the parabolic equations by using the Green's functions. We consider Dirichlet-Neumann and Neumann-Robin boundary conditions for the substrates equations and their extension to the case with variable
Keywords
Cite
@article{arxiv.2411.17974,
title = {Existence and uniqueness of solution to a hyperbolic-parabolic free boundary problem for biofilm growth},
author = {Dieudonné Zirhumanana Balike and Luigi Frunzo and Maria Rosaria Mattei and Fabiana Russo},
journal= {arXiv preprint arXiv:2411.17974},
year = {2025}
}
Comments
We kindly request the withdrawal of our submission due to a misunderstanding among the co-authors regarding the decision to upload the work to arXiv. Not all authors had agreed to its public release, and the submission was made based on an internal miscommunication. We regret this oversight and appreciate your assistance in addressing this matter